Invariants to affine transform
What is affine transform?
u  a1 x  a2 y  a0
v  b1 x  b2 y  b0
Why is affine transform important?
• Affine transform is a good approximation of projective
transform
• Projective transform describes a perspective
projection of 3-D objects onto 2-D plane by a central
camera
Affine moment invariants
Many ways how to derive them
• Theory of algebraic invariants (Hilbert, Schur,
Gurewich)
• Tensor algebra, Group theory (Lenz, Meer)
• Algebraic invariants revised (Reiss, Flusser & Suk,
Mamistvalov)
• Image normalization (Rothe et al.)
• Graph theory (Flusser & Suk)
• Hybrid approaches
All methods lead to the same invariants
General construction of Affine Moment Invariants
General construction of Affine Moment Invariants
Affine Moment Invariants
Simple examples of the AMI’s
Graph representation of the AMI’s
Graph representation of the AMI’s
Graph representation of the AMI’s
Graph representation of the AMI’s
Graph representation of the AMI’s
Removing dependency
Affine invariants via normalization
Many possibilities how to define normalization
constraints
Several possible decompositions of affine
transform
Decomposition of the affine transform
• Horizontal and vertical
translation
• Scaling
• First rotation
• Stretching
• Second rotation
• Mirror reflection
Normalization to partial transforms
• Horizontal and vertical translation -m01 = m10 = 0
• Scaling -- c00 = 1
• First rotation -- c20 real and positive
• Stretching -- c20 =0 (μ20=μ02)
• Second rotation -- -- c21 real and positive
Properties of the AMI’s
Application of the AMI’s
• Recognition of distorted shapes
• Image registration
Landmark-based robot navigation
Clusters in the space of the AMI’s
Image registration
Landsat TM
SPOT
Selected regions
Matching pairs
Image registration
Region matching by the AMI’s
Robustness of the AMI’s to distortions
Robustness of the AMI’s to distortions
Aspect-ratio invariants
Projective moment invariants
Projective transform describes a perspective projection
of 3-D objects onto 2-D plane by a central camera
Projective moment invariants
• Do not exist using any finite set of moments
• Do not exist using infinite set of (all) moments
• Exist formally as infinite series of moments of both
positive and negative indexes
Invariants to contrast changes